Am I using too much weight (workout)?

This is quite normal. You’re just running out of energy. Consider bringing something to eat that is rich in carbohydrates. (Even sugar is quite okay during a workout session.)
Also consider taking longer breaks in between exercises.

Come to think of it, you could combine them and make it a snack break. I’d also advise eating something rich in protein during your workout session. (I use bits of chicken breast for this. There are many alternatives.)

Do use as much weight as you can handle without injuring yourself.

I don’t know for sure but when I was in my P.T. class we did 3 different upper body lifts one day and 3 lower body lifts the next day and we just kept on alternating

As @Fyrius said, it sounds perfectly normal. A good rule I’ve used is to find your maximum one-rep weight, then use 80% of that weight for your workout, e.g. if you can lift 100 lbs one time, but struggle too much to properly lift it a second time, use 80 lbs for that workout. You should be able to do about 3 sets of 8–14 reps with the 80 lbs. Determine your new one-rep max periodically.

I’m not a fan of machines, I like free-weights much better as you know exactly how much weight you’re lifting and you’re lifting it without any assistance… but, you didn’t ask for my opinion.

Anyway, yes, it is normal. If you’re able to get 12 reps in your first set, add a little more weight. If by the last set you’re only getting 8 reps in, the weight you’re set at is probably perfect.

@Disc2021: Seconded.
From what I’ve seen, machines usually only offer lighter versions of exercises you could do perfectly well with your body weight. The machines I’ve seen just emulated pull-ups, dips, broad push-ups, squats and glute/ham raises.
I suppose it might be a good place to start, if you can’t do it with your body weight. But beyond that I agree, free weights are superior.

Since your goal is to gain strength you should be doing 6–12 reps per set, 2–5 sets total. 3 sets is definitely sufficient per exercise. You should first work the bigger muscle group, in this case you chest workout with 3 exercises, 2–5 sets per exercise i.e. 3 sets flat bench, 3 sets incline, 3 sets flyes. Then you should move to your tricep workout following the same pattern. Idk how much you weigh or how long you have been lifting for so I cannot give you an exact amount of weight that you should be lifting. If you are dead by the time you are almost done with your final set in your last exercise, you are probably lifting too much. Going slow and getting a good pump with less weight is great. Try getting the right form down and focusing on your technique. If you find that it is too easy and you feel like you are not getting enough from it, you can try adding 5–10% more weight each set. Do a lot of experimenting and find what is right for you….everyone is different.

Also, depending on your age, you should probably be using free weights. Free weights being dumbbells and barbells. Multijointed exercises are best because they work the surrounding muscles, which leads to bigger gains for areas like the chest because it will work your shoulders, triceps thus making your chest look bigger. If you are older, machines are fine. You will see better results if you use barbells and dumbbells however, so I would definitely suggest using those more if you are capable